You don;t need to use the paralist package at all. Just use the starred variant of the environments from the enumitem package -- requires to add the inline option to the enumitem package:

If you don't want the enumerat in the tabular to be on its own line you can tweak all the paramaters used for vertical spacing. Below I used [topsep=0pt,partopsep=0pt,itemsep=0pt, parsep=0pt] to achieve:

References:
Compilation Error
You original compilation error can be fixed via:
- Load
paralist before enumitem.
- Replace the
\end{cenum} with \end{compactenum}
Code: inline
\documentclass{article}
\usepackage[inline]{enumitem}
%\usepackage{paralist}
\begin{document}
\begin{tabular}{lp{10cm}}
\scshape Assume: & \begin{enumerate*}
\item $(p_1, p_2)$ is an optimal price sequence
\item $p_2 > p_1 > 0$
\end{enumerate*}
\\
\scshape Prove: & False
\end{tabular}
\begin{enumerate}[label=\bfseries Case \arabic*:,leftmargin=*]
\item
It rains today.
\item
It does not rain today.
\end{enumerate}
\end{document}
Code: Tight Spacing:
\documentclass{article}
\usepackage{enumitem}
%\usepackage{paralist}
\begin{document}
\begin{tabular}{lp{10cm}}
\scshape Assume: & \begin{enumerate}[topsep=0pt,partopsep=0pt,itemsep=0pt, parsep=0pt]
\item $(p_1, p_2)$ is an optimal price sequence
\item $p_2 > p_1 > 0$
\end{enumerate}
\\
\scshape Prove: & False
\end{tabular}
\begin{enumerate}[label=\bfseries Case \arabic*:,leftmargin=*]
\item
It rains today.
\item
It does not rain today.
\end{enumerate}
\end{document}
Code: Fix Compilation Error
\documentclass{article}
\usepackage{paralist}
\usepackage{enumitem}
\begin{document}
\begin{tabular}{lp{10cm}}
\scshape Assume: & \begin{compactenum}
\item $(p_1, p_2)$ is an optimal price sequence
\item $p_2 > p_1 > 0$
\end{compactenum}
\\
\scshape Prove: & False
\end{tabular}
\begin{enumerate}[label={\bfseries Case \arabic*:},leftmargin=*]
\item
It rains today.
\item
It does not rain today.
\end{enumerate}
\end{document}